If log2(3𝑥 + 𝑦) = 1 and log2 (

Question

If log2(3𝑥 + 𝑦) = 1 and log2 (

𝑥
𝑦
) = −2, find
the values 𝑜𝑓 𝑥 𝑎𝑛𝑑 𝑦
A. 𝑥 = 8
5 , 𝑦 = −32
B. 𝑥 = 2
7 , 𝑦 = 8
7
C. 𝑥 = −2, 𝑦 = 8
D. 𝑥 = 1, 𝑦 = 4

Answer 1

We can use the properties of logarithms to rewrite the given expressions in a more useful form. For example, from the first equation, we have:

log2(3𝑥 + 𝑦) = 1
2^1 = 3𝑥 + 𝑦
1 = 3𝑥 + 𝑦

Similarly, from the second equation, we have:

log2(𝑥/𝑦) = -2
2^-2 = 𝑥/𝑦
1/4 = 𝑥/𝑦
𝑥 = (1/4)𝑦

Now we can substitute this expression for 𝑥 into the first equation and solve for 𝑦:

1 = 3𝑥 + 𝑦
1 = 3(1/4)𝑦 + 𝑦
1 = (3/4)𝑦 + 𝑦
1 = (7/4)𝑦
𝑦 = 4/7

Substituting this value back into the expression for 𝑥, we get:

𝑥 = (1/4)𝑦
𝑥 = (1/4)(4/7)
𝑥 = 1/7

Therefore, the answer is B: 𝑥 = 2/7, 𝑦 = 4/7.

Answer 2

To solve for the values of x and y, we'll use the properties of logarithms.

First, let's solve for x and y separately.

1. Solving for x:
We have: log2(3x + y) = 1

By the definition of logarithms, this equation can be rewritten as: 2^1 = 3x + y
Simplifying: 2 = 3x + y

2. Solving for y:
We have: log2 (xy) = -2

By the definition of logarithms, this equation can be rewritten as: 2^(-2) = xy
Simplifying: 1/4 = xy

Now, we have two equations:
1) 2 = 3x + y
2) 1/4 = xy

To find the values of x and y, we can solve this system of equations.

Multiplying equation 2 by 4, we get: 1 = 4xy

Now we have the equations:
1) 2 = 3x + y
2) 1 = 4xy

Rearranging equation 1, we get: y = 2 - 3x

Substituting y in equation 2, we get: 1 = 4x(2 - 3x)

Expanding and simplifying: 1 = 8x - 12x^2

Rearranging this quadratic equation: 12x^2 - 8x + 1 = 0

To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 12, b = -8, and c = 1.

Let's solve for x using the quadratic formula:

x = (-(-8) ± √((-8)^2 - 4 * 12 * 1)) / (2 * 12)
x = (8 ± √(64 - 48)) / 24
x = (8 ± √16) / 24
x = (8 ± 4) / 24

This gives us two possible values for x:
1) x = 12/24 = 1/2
2) x = 4/24 = 1/6

Next, let's substitute these values of x into the equation y = 2 - 3x to find the corresponding values of y.

For x = 1/2:
y = 2 - 3(1/2) = 2 - 3/2 = 1/2

For x = 1/6:
y = 2 - 3(1/6) = 2 - 1/2 = 3/2

Therefore, the values of x and y are:
A. x = 1/2, y = 1/2
B. x = 1/6, y = 3/2

So, none of the given answer choices are correct.